The area under tree canopies allows little sun and water to get through to lawn grass. Grass also has a very tough time competing with tree roots for nutrients from the soil. One option is to get a tree professional to remove some lower tree limbs or thin the crowns of the trees to let more light in. There's still no guarantee that grass will get established.
Shade-tolerant grass varieties are available, but remember the key word is tolerant — no grass is likely to grow well for long in deep shade. As far as grass types for shade, St. Augustine is a good warm-season variety; red fescue or Chewings fescue are good cool-season types. Sun / shade seed blends are also available. Plant warm-season grasses in early spring; cool-season grasses in fall.
When grass seed is planted, provide a lot of water to get the root system established. Make sure you mow it as high as possible. Some types will require regular reseeding. High-nitrogen fertilizer promotes blade growth, which is not really what you want. Look for a fertilizer with a higher concentration of phosphorus to promote root growth. Before undertaking any major fertilizer program, get a soil test kit and follow the recommendations.